A height the map already knows

Everything about satellite navigation on the Earth assumes a constellation built for the purpose: thirty satellites in medium orbits, spread so that at every place and moment at least four are in view and their geometry is good. The Moon has nothing of the kind. Spacecraft there are navigated from the Earth, by the Doppler shift and delay of radio signals — one direction of measurement from one place, rescued only by watching the spacecraft move for hours or days. That is adequate for an orbiter. It is not adequate for a rover that needs to know where it is now, or for a lander descending into a crater near the south pole, where the water ice and the long-lit ridges make the most valuable real estate on the Moon.

The plans for lunar navigation services propose a handful of satellites, not thirty, and they make up for the small number by choosing their orbits carefully. A satellite on an elliptical orbit spends most of its time near the far end, moving slowly; put the far end over the south pole and the satellite hangs above the polar region for nine hours of every thirteen. A few such satellites, phased so that their slow apolune passes overlap, could keep the pole served most of the time.

The hero figure shows what six of them give a receiver at the pole. Each satellite has its own orbital plane, the planes’ nodes spaced evenly round the Moon; every orbit has a semi-major axis of 6,541 kilometres, an eccentricity of 0.6 and an inclination of 56.2°, a combination chosen because lunar gravity leaves such an orbit’s shape and apolune latitude nearly unchanged for years — a frozen orbit. The horizontal dilution of precision, the factor by which the geometry scales range errors into horizontal position errors, is good whenever a fix is possible: a median of 1.7. The vertical is not: a median of 6, and peaks of twenty. And for more than a quarter of the time fewer than four satellites are above a 5° horizon at all.

Orbits that hold their shape #

Lunar orbits are not stable in general. The Moon’s gravity field is lumpy — mass concentrations under the large basins pull low orbits out of shape within months — and the Earth’s tide, ten times stronger relative to lunar gravity than the Sun’s is on the Earth’s satellites, slowly trades an orbit’s eccentricity against its inclination in the long cycle that turns an inclination into an eccentricity. A satellite left in an arbitrary lunar orbit drifts until its perilune meets the surface.

A frozen orbit is one placed at a fixed point of that slow exchange. For particular combinations of inclination, eccentricity and argument of perilune the averaged effects of the Earth’s tide and the Moon’s own flattening cancel, so the orbit’s shape and the latitude of its apolune stay nearly constant for years with little or no correction. The elliptical frozen orbits proposed for lunar navigation are of this kind: their apolune stays over the south pole because the dynamics keep it there, and the constellation needs station-keeping only to hold the satellites’ relative phases.

That is what makes a small constellation viable at all. A satellite whose apolune drifted from the pole by a few degrees a month would serve the pole for a season and then serve somewhere else. The frozen orbit fixes the service region to the pole for the mission’s life, and it fixes, by the same token, the cone that the satellites form above it.

Four satellites on a cone #

The vertical geometry is bad for a reason that has nothing to do with how many satellites there are. A navigation receiver solves for four unknowns: three components of its position and the offset of its own clock from the satellites’ time. Each range measurement constrains a combination of them. A change in the receiver’s clock lengthens every range by the same amount; a change in its height lengthens each range by the sine of that satellite’s elevation. If all the satellites stand at the same elevation, those two changes are indistinguishable — a receiver lower by some distance and a receiver whose clock is late by the corresponding time produce identical ranges — and the solution cannot separate them.

On the Earth this degeneracy is broken by satellites at every elevation: one nearly overhead, others near the horizon, the rest between. The vertical is still worse than the horizontal — every satellite is above, none below — but the clock and the height can be told apart. Above the Moon’s south pole the polar constellation does the opposite. Its satellites near apolune are all at about the same distance and the same latitude, so seen from the pole they stand at nearly the same elevation, around fifty degrees, spread round the sky in azimuth: a cone. The design that makes the constellation efficient — every satellite parked at the same place in its orbit, high over the pole — is the design that puts them on the cone.

What four satellites give #

The minimum constellation is four, and it is worth seeing what four give before seeing how to do better.

With four satellites the service is mostly gaps. Each satellite is in view of the pole for about two-thirds of its orbit, and a fix needs all four at once; with their phases spread over half an orbit, all four are up together for only two-fifths of the time. Spreading the phases further makes the gaps shorter but more frequent, and bunching them makes the windows longer but the gaps longer too: with exactly four satellites the constellation is a scheduling problem in which the four ranges must line up. Within the windows the geometry is the cone’s: excellent horizontally, poor vertically.

A height the map already knows #

A receiver on the surface of the Moon has a piece of information a receiver in an aircraft does not: it is on the surface. Its height above the Moon’s centre is the height of the terrain where it stands and the terrain of the polar regions has been mapped by laser altimetry from orbit to a few metres vertically, at a horizontal resolution of tens of metres. That height can be fed to the navigation solution as if it were one more measurement — a constraint on the vertical coordinate with an uncertainty set by the map.

Its effect is out of proportion to its simplicity. A height measurement constrains the vertical coordinate directly, without involving the clock, which is exactly the direction the cone cannot resolve. And it counts as an equation: with the height known, three satellite ranges suffice to solve for the two horizontal coordinates and the clock.

The two curves do not differ by a small correction. With ranges alone, eight satellites give a usable fix less than half the time, because even when enough are in view the cone spoils the vertical. With the height added, four satellites do better than eight did, and six give a fix all the time. Three satellites, which can never give a fix from ranges alone, give one forty-four per cent of the time.

The weighting matters less than might be expected, up to a point. Recomputed with the height five times less certain than a range, the aided availability is unchanged — 65 per cent with four satellites, all the time with six — because what the cone lacks is not precision in the vertical but any vertical information independent of the clock, and even a coarse height supplies it. At ten times less certain the benefit collapses on this criterion, to 19 and 38 per cent: the height can no longer pull the vertical dilution below the threshold. With range errors of a metre or two, that edge falls at a terrain uncertainty of ten to twenty metres, which the polar maps beat almost everywhere outside the shadows.

The map is worth more than two satellites. That is the arithmetic of the figure, and it holds for a reason: the map supplies exactly the measurement the constellation’s geometry cannot, and a satellite added to the cone supplies more of the measurement the geometry already has.

A crater is a horizon mask #

The most valuable sites near the south pole are in or beside craters, and a crater’s rim hides the low part of the sky. A receiver inside one sees only satellites above the rim — an elevation mask of fifteen degrees or more.

The mask hurts both curves, and the ranges-alone curve more: the satellites it removes are the lowest, the only ones that could have broken the cone from below. With the terrain height the service survives the mask far better — six satellites still give a fix 85 per cent of the time inside a fifteen-degree rim, and eight all the time — because the receiver no longer depends on the low satellites for its vertical.

The polar craters have a further twist. The interiors of the permanently shadowed ones, where the ice is, are mapped less well than the sunlit terrain, because the laser altimeters measured them but the imaging that pins down horizontal positions could not see into them. The height a receiver there would use comes with a larger uncertainty, and the aid it provides is correspondingly weaker. The places where navigation would be hardest are the places where the map that rescues it is least certain.

The rim also shortens every satellite’s pass. A satellite’s time in view is the time it spends above the mask, and the circle of sky a station can see shrinks quickly as the mask rises: for satellites near apolune, which cross the sky slowly at fifty degrees, a fifteen-degree rim costs little; for satellites on the way in or out of perilune, which cross low and fast, it costs most of their pass. Those passes were the ones that supplied low-elevation geometry, and their loss is why the ranges-alone curve suffers more than the aided one.

A lander choosing its site can trade among these effects. A site on a crater rim sees the whole sky and the full constellation; a site on the floor sees less but may be where the science is. With the terrain height in the solution the difference between the two is tens of per cent of availability rather than the difference between a service and none, which turns site selection from a navigation constraint into a scientific choice.

Service that falls away from the pole #

The constellation is aimed at one pole, and the service shows it.

Down to about forty degrees south the service is good, because the satellites’ long apolune passes are visible from the whole southern polar cap. Beyond that it falls steeply: from the equator the satellites near apolune are low in the southern sky or below the horizon, and the ones near perilune pass quickly. The northern hemisphere has almost nothing. A small constellation is a regional service, and the region is set by the latitude of its apolune — which is why the plans for lunar navigation start with the south pole, where the first crewed landings are to be, and add satellites for other regions later.

Why the Earth’s systems never needed this #

Terrestrial navigation receivers can use the same trick, and some do. Aircraft receivers can be aided by a barometric altimeter; a receiver at sea knows it is at sea level. But on the Earth the aid is a convenience, used when satellites are blocked, because the full constellations give good vertical geometry most of the time. Five satellites can cover the whole Earth only from high orbits and only just; the operational systems carry thirty or more, partly to ensure that low and high satellites are always in view together.

On the Moon, with four to eight satellites and every one of them parked over the same pole, the aid is not a convenience but a requirement. The comparison is a lesson about where information is cheapest. Launching a navigation satellite to lunar orbit costs hundreds of millions; the polar terrain has already been mapped by spacecraft that were sent for other reasons. A navigation design that treats the terrain model as part of the system rather than as a backup gets more service from fewer satellites, and the trade can be stated in satellites: on these numbers, one map is worth two or more spacecraft.

Where the satellites’ time comes from #

A navigation fix is a measurement of time: each range is a travel time multiplied by the speed of light, and a nanosecond is thirty centimetres. The satellites’ clocks have to agree with each other to a few nanoseconds and their positions have to be known to a metre, and on the Earth both are maintained by a ground network that tracks every satellite continuously. A lunar constellation’s satellites are tracked from the Earth, 380,000 kilometres away, one direction at a time, and possibly from each other.

The clocks bring a complication of their own. A clock on the Moon’s surface runs faster than one on the Earth’s by about fifty-six microseconds a day, because it sits higher in the combined gravitational potential and moves differently, and a clock on a satellite in an eccentric lunar orbit changes its rate round every orbit as its height in the potential and its speed change. A lunar navigation service needs a time scale of its own, defined for the Moon’s surface and related to the Earth’s by a formula with periodic terms, and proposals for exactly that are under discussion — a seventh entry for the list of kinds of second that astronomy already keeps.

None of this changes the geometry, but it is why the receiver’s own clock is treated as unknown in every calculation here. A satellite clock steered from the Earth, read through a relativistic correction, by a receiver whose own quartz oscillator drifts by microseconds a second, gives no basis for assuming the receiver knows the time. The clock is the fourth unknown because it has to be, and the cone makes the fourth unknown expensive.

Where the model stops #

The orbits are Keplerian. Real frozen orbits keep their shape on average, through a balance of the Moon’s lumpy gravity and the Earth’s tide, but oscillate about it; apolune altitude and latitude wander by a few per cent over weeks. The coverage averages are insensitive to that; the timing of particular gaps is not.

The height is weighted like one range. A terrain model good to a few metres is as good as a typical range measurement, which is the weighting used. Where the map is worse — in permanently shadowed craters — the aid is weaker, and the curves fall toward the ranges-alone ones.

Every satellite’s clock and position are perfect. The service’s actual accuracy is the dilution of precision times the error in each range, which includes the satellites’ own orbit and clock errors as broadcast. For a small constellation navigated from the Earth those errors are larger than for the terrestrial systems, and they multiply the dilutions drawn here.

The receiver’s clock is free. A receiver with a very stable clock — an atomic clock on a lander — could constrain the clock the way the map constrains the height, and would break the cone the same way. The map is used here because every surface receiver has one and few carry atomic clocks.

Still open: navigating in the dark #

The receivers that need this service most will be in the permanently shadowed craters: rovers prospecting for ice, where no starlight lets a camera match the surface to a map and no Sun lights a horizon to sight against. There the satellites are the only reference, the crater rim is the mask, and the terrain map is at its least certain. Whether a few satellites and a map of the shadows can give a rover a position good to the metres it needs to return to a drill site — and how many satellites it takes once the map’s errors in the dark are counted honestly — is the question the first constellation will answer, and the answer will be measured in the crater rather than computed above it.